270 
PROFESSOR M. J. M. HILL OH THE LOCUS OF SINGULAR POINTS 
(III.) When the fundamental equations cannot be satisfied by any values of the 
coordinates which make the discriminant or a factor of it vanish, the values of the 
parameters being finite. 
(IV.) When the three fundamental equations, which contain the five quantities 
x, y, z, a, b are equivalent to two relations only between them. 
(V.) When the three fundamental equations, which contain the five quantities 
x, y, z, a, b are equivalent to one relation only between them. 
I. The Fundamental Equations are satisfied by values of the coordinates which are 
functions of one 'parameter only. 
In this case, eliminating the parameter, two relations between the coordinates are 
obtained. Hence the locus of ultimate intersections is a curve. 
Example 18. 
Let the surfaces be 
era? 3 — 2 abxy + bfi 9, — 2a (a? + 1) — 2 by + 2=0 . . . (197). 
(A.) The Discriminant. 
It is 
a? 3 — xy — a? — 1 
- xy if -y 
— x — 1 — y z 
— — if (2a? + l) 3 . 
(B.) The coordinates of each point on the locus of ultimate intersections must 
satisfy (197) and 
a/jr 
bxy — (a? + 1) = 0 
(198). 
From (197) and (198) 
— axy + by 1 — y = 0 
a (a? + 1) — by + 2 = 0.(199). 
(i.) Now a solution of the second of equations (198) is 
y = 0 .(200). 
Substituting in the first of equations (198) and in (199) 
aa? 3 — (a? + 1) = 0.(201), 
— a (a? + 1) + 2 = 0 .(202). 
